FkBKUAKV 2 2, 1894] 



MA TURE 



38; 



LETTERS TO THE EDITOR. 



[The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond -with the writers of, rejected 

 manuscripts intended for this or any other part <?/ Nature. 

 No notice is taken of anonymous communications.'^ 



On M. Mercadier's Test of the Relative Validity of 

 the Electrostatic and Electromagnetic Systems of 

 Dimensions. 



A SERIES of papers, by M. Mercadier, on the dimensions of 

 physical quantities, has recently bren appearing in the Comptes 

 Rendtts. They are summarised and extended in the Journal 

 de Physique (July, 1893, p. 289). 



In a note (p. 296) the author stales that — "En 1883 nous 

 avons montre, M Vaschy et moi, separement d'ahord, puis en 

 •collaboration, que les deux systemes [of the dimensions of elec- 

 trical and magnetic quantities] imagines par Maxwell etaient 

 contradictoires et que I'un d'eux e'ait inadmissible. Dcpnis, en 

 particulier MM. Hertz en 1885 {IVied. Ann. t. xxiv. 1885) et 

 Riicker {Phil. Mag. 5*^ serie, t. xxvii, 1889) sans mentionner 

 notre travail, sont arrives aux memes conclusions." 



In general it is, I think, wisest to leave such claims to 

 priority alone, but as M. Mercadier's paper has appeared in two 

 important French journals, I should like to make a few remarks 

 on the history of the dimensional formulae of electrostatic and 

 electromagnetic quantities, which I do the more readily because 

 I have no claims to priority to establish on my own behalf. 



Maxwell's theory leads to the conclusion that between elec- 

 trical, magnetic, and ordinary dynamical quantitie-; there exi-t 

 relations which are one less in number than the electrical and 

 magnetic unknowns. Hence the dimensions of all the latter 

 can be expressed in terms of length, mass, time, and of the 

 unknown dimensions of any one of ihem. Maxwell gave two 

 examples of such expressions in which electrical quantity and 

 strength of magnetic pole are the unknowns selected (vol. ii. 

 first ed. p. 241). It is absurd to suppose that he did not know 

 that similar tables could be drawn up in terms of specific 

 inductive capacity (K) and magiietic permeability {fj.). 



After discussing the general problem, Maxwell expressefi the 

 opinion that the "only systems of any scientific value are the 

 ■electrostatic and electromagnetic systems" {loc. cit. p. 241), and 

 proceeded to explain how they are obtained. Unfortunately, 

 in order to emphasise the fact that the fundamental assumption 

 in the electrostatic system is that the specific inductive capacity 

 of the standard medium (air) is taken as unity, he used a nota- 

 tion in which K is represented as without dimensions instead as 

 of unknown dimensions. 



It must also, I think, be admitted that this notation con- 

 duced to the use of phrases which mis^ht very easily mislead. 

 Thus {loc. cit. pp. 368-9) he concludes that a quantity n, 

 which he defines as " the number of electrostatic units in one 

 electromagnetic unit," " is a velocity." Had the symbols K 

 and /u been retained, his argument would have led to the con- 

 clusion that n [fjL-h K-'>] is a velocity. The matter is not of 

 the first importance, but a notation which requires the state- 

 ment that the ratio of two like things "is a velocity," makes 

 the subject unnecessarily difficult. 



Any doubt to which such expressions might have given rise 

 has, however, been completely set at rest since 1882. 



A discussion, initiated by Prof. Clausius, then took place 

 in the Philosophical JlJagazine ($\h series, vols. xiii. and xiv.), 

 in which Profs. Everett, J. J. Thomson, Lodge, and Larmor 

 took part. It is only necessary for my present purpose to cite 

 the fact that Prof. Lodge explicitly stated that "the number of 

 fundamental relations must be limited by the number of 

 fundamental experiments, viz. three — Coulomb, Coulomb, and 

 Oersted ; and the shortest way of writing the independent rela- 

 tions is this : — 



[fie"-] = [Km"-] = [ML] 

 and 



[f^Kv-]=i. 



The electrostatic convention makes [K] = i ; the electro- 

 magnetic convention makes [/x] — i." 



This paper was published in September 1882. 



In January 1883 a paper by MM. Mercadier and Vaschy 

 appeared in the Comptes Rendus. No reference was made by 

 them to the discussion between English and German physicists, 



NO. I 269, VOL. 49] 



but up to a certain point they adopted precisely the same line 

 of argument as that with which we were familiar in England. 

 Taking the formulae 



and 



f=^kn 



r Jii'dsds' 



f OC k ;; 



they had no difficulty in showing that 



m = m 



It was well known in 1883 that Maxwell's theory requires 

 only two constants, K and n, to define the constitution of the 

 medium, and that 



L^] = 



[R].f"'= 



[m] 



(Maxwell, vol. ii. p. 289, equation 24). 



Hence from this point of view the equation 



is the same as 



m - m 



[fiKv^] = I. 



Up to this point, therefore, there was nothing in the paper of 

 MM. Mercadier and Vaschy which could not be directly deduced 

 by Maxwell's theory from the explicit statements ol Lodge. 



Alter this they proceeded to develop the sufiject further in an 

 argument which may be summarised as follows : — 



The constant k is inversely proportional to the specific in- 

 ductive capacity. 



Specific inductive capacity is proportional to the square of 

 the index of refraction. 



The index of refraction is inversely proportional to the velocity 

 of light in the medium. 



Hence k = aV", "o etait une constant numerique" (the 

 italics are in the original). 



Hence, since 



[1] = [V-] and [k] = [V-], 



k' is a number "et I'emploi du systeme electromagnetique 

 d'unite- electriques se trouveraii justifie thhriquement." 



The fallacy in this is obvious even if the experimental justifi- 

 cation of the siep k' ca (lefractive index)- be admiiti-d. 



Because k (x V- it <ioes not follow that [^] = [V^ unless we 

 are sure that all the physical conditions have lieen included in 

 the equal ion. Yet Maxwell had given the strongest reason to 

 believe that the magnetic permeability was also involved. He had 

 distinctly pointed out that K would only vary as the square of 

 the refractive index if fx were cnstant (vol. ii. p. 388). In 

 other words, M Mercadier, arguing from experiments on ma- 

 terials whose magnetic permeabilities differ bui little from that of 

 air, treats [/J'] = [V'-J or [K] = [V"^] as an independent equation. 

 He thus assumes that k' or fx. is of no dimensio s, and then proves 

 the truth of the electromagnetic system which i- avowedly based 

 on that assumption. It would be difficult to find a more com- 

 plete instance of arguing in a ciicle. 



The point, h >wever, is elaborated by experiment and further 

 argument in another pai'er {C. R. t. xcvi. 1883, p. 250). 



The conclusion that k' is a constant is supported by the 

 fact that an induced current is arteris paribus the same, whether 

 the currents are or are not surrounded by non-magnetic materials 

 such as alcohol and i>eiizene. 



Again the statement is made — ten years after the publication 

 of Maxwell's book — " D'apres les idees universellement ad- 

 mises, les coefficients des formules de magreiisme et d'electro- 

 magnetisme seront analogues a. k' par consequent ils devraient 

 eire comme lui independants des milieux " (p. 252). 



The fact is that, according to Maxwell's theory, one of these 

 coefficients is independent of the medium, while the other two 

 vary, the one inversely as the other, when the medium is 

 changed. 



Quite apart then from the question as to whether these factors 

 rep'C enied pure numbers or concrete quantities, ii was at that 

 date almost universally believed that the values of two of them 



